Silver L X-ray energies were calculated using theoretical binding energies from relaxed orbital relativistic Hartree-Fock-Slater calculations. Theoretical X-ray energies are compared with experimental results

Aoyagi, M.

Chen, M. H.

Crasemann, B.

Mark, H.

English

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	 Interpretation of the silver L x-ray spectrum*
''.	 Mau Hsiung Chen and Bernd Crasemann3.._:
Departmeirt of Physics, University of Oregon
Eugene, Oregon 97403
and
Michio Aoyagi and Hans Mark
Ames Research Center, NASA
Moffett Field, California 94035
A new analysis of the Ag L-series x-ray spectrum has been performed, based
on relaxed-orbital relativistic transition-energy calculations. Satellite
energies were computed in intermediate coupling. It is found that satellites
arising from IN double-hole states generally fall within the natural width
of the parent diagram lines. Contrary to previous assumption, the observed
high-energy satellites are due to LM double-hole states produced by L1
-L3M4 5r
Coster-Kronig transitions and by shakeoff. The observed peak structure in
the satellite spectrum is due to multiplet splitting of the initial and final
double-hole states. Theoretical L 1-L3M4 , 5 transition rates based on the new
Coster-Kronig energies are closer to experiment than previous results, but
still are too large by a factor of 'u2, indicating an as yet undetected flaw in
the theory.
l
*Work supported in part by the U.S. Army Research Office (Grant DAHC04-75 -G-
0021) and by the National Aeronautics and Space Administration (Grant
NGR 38-003-036).
(
2I. INTRODUCTION
The Ag L x-ray spectrum has recently been analyzed by McGuire, 1
 on the
basis of Parratt's measurements. 2 McGuire used calculated radiative, Coster-
Kronig, and Auger transition rates and a hypothesis on the origin of L-series
x-ray satellite lines to compute the Ag L x-ray spectrum. In order to obtain
reasonable agreement between theory and experiment, McGuire proposed that
LL L3M4 5 Coster-Kronig transitions are energetically impossible in Ag and
that the L satellites arise from IN doubly ionized states.
A systematic ab initio computation of L Coster-Kronig transition energies
has now been carried out by Chen et al., 3 who used an improved relativistic
Hartree-cock-Slater potential model. 4 These calculations show that, while
L  L2M4 5 transitions are energetically forbidden in Ag, this. is not the case
r
for  L1
 L3M4 5 transitions, contrary to McGuire's I assumption: the LL L3M4
,
transition energy is found to be 31 eV, and the LL L 3M5 energy, 39 eV.
In the light of the:_ new Coster-Kronig energy calculations, we proceed
to re-analyze the Ag L x-ray spectrum and to re-examine the origin of the
satellites. In Sec. II, we compare theoretical and experimental x-ray
energies. In Sec. III, we describe a calculation of the x-ray spectrum and
compare-$he results with experiment.
II. L X-RAY ENERGIES
Silver L x-ray energies were calculated using theoretical binding energies
from relaxed-orbital relativistic Hartree-Pock-Slater calculations.4
Theoretical x-ray energies are compared in Table I with experimental results. 2,5
Agreement between calculated and measured energies is reasonably good, within
i
u'
37 eV, showing that the theoretical transition -energy calculations are good to
within an uncertainty of a few ev.
The same relativistic Hartree-Fock -Slater model has been u p^d to calculate
the x-ray energy shift between diagram and satellite lines (Table II). It is
found that an Ni spectator hole causes the Lal 2 and L51 x rays to shift by
50.5 eV. The measured La1,2 _and  Lpl x-ray widths  are '.2.3 eV, hence the
satellites of these lines arising from IN  double ionization will be indis-
tinguishable from the diagram lines. Similarly, satellites arising from N-
shell spectator vacancies will also coalesce with other L-M transition diagram
lines, because these lines are much wider than La 1 2 and LO 1,2 Even L$ (L 3-N5)
satellites produced by an extra N-shell vacancy will disappear within the
natural diagram-line width: the N. hole shifts the LO  transition energy by
S5 eV (Table II), which falls within the observed 2 x-ray width.
The Lal 2 satellites observed by Parratt 2 are shifted from 10 to 27 eV
with respect to the diagram line, whence they cannot be due to IN doubly
ionized atoms. An additional M1 hole, on the other hand, shifts La l 2 and LS 
transitions by 1b10 eV and LR 2 transitions by ti33 eV, which is of the same
order of magnitude as the observed satellite shifts. We conclude that
Parratt's 2 Ag L sateilites arise from LM doubly ionized atoms, and not from
IN atoms.
We have calculated the energies of the Lal 2 satellites that arise from
L3M4,5 M4,SM4,5 transitions, using intermediate coupling  and the relativistic
Hartree-Fock-Slater model. 3 ' 4 The relativistic LS multiplet energies were
obtained by following Larkins' approach. 7 Results are listed in Table III.
The terms indicated in the first column of Table III are the dominant com-
ponents in the wave-function expansion. The eight initial states included in
i
4Table III are the states that can be reached by L L L3M4 5 Coster-Kronig
r
transitions in Ag. For comparison, the observed broad satellite peaks are
also included in Table III. These peaks can be interpreted as resulting from
many closely bunched lines. The gross structure of the observed peaks agrees
quite well with the calculation.
III. RELATIVE INTENSITIES
A. Theoretical model
In treating the relative intensities, we have followed the model of
Krause et al. S
 We take into account (a) the initial L-hole population
N1 , N2 , N3 and shakeoff probabilities P 1 , P2, P 3 ; (b) shifting of the L-shell
holes by Coster-Kronig transitions, with yields f12 , f13, f23 ; (c) distinc-
tion between satellite lines that arise from LM and IN double-hole states.
In order to separate the LM and LN double-hole states, we express the
shakeoff probabilities and Coster-Kronig yields as
Pi
 - PiM + PiN	 (1)
and
f13 = fi3M + fijN,
where Pix is the X-shell shakeoff probability during i-shell ionization, and
f.	 is the Coster-Kronig yield leading to the'jX double-hole final state.
The number of single-hole states left after Coster-Kronig transitions
have taken place is
n1(L1) 	 (1-P1)(1-f12-f13)N1
n2 (L2 ) 	 (1-P 2 ) (1-f23)NZ	(3)
.	 a
n3(L3)	 (1-P3)N3.
I	 I^	 I	 I	 I	 ',
5
The number of double-hole states is
n1 (L1M) = P1MN1(l-fl2-fl3)
n1 (L1N) = P1NN1(1-f12-f13)
n2 (L2M) = P2MN 2 (1-f 23)
(4)
n2 (L2N) = P2NN2(1-f23) + f12(1-P1)N1
n3(L3M)
	
P 3MN3 + fl3M(1-21)N1
n3 (L3N) = P3NN3 + f23 (1-p2 )N2 + f13N(1-P1)NV
Here, we have used the fact that fl2
=f 12N and f23 f23N'
We neglect the contribution due to triple-hole states.
In Sec. II we concluded that the satellites arising from IN double-hole
states are not resolved from diagram lines, and that the observed satellite
i
structure is due to LM doubly ionized atoms. The number of hole states pro-
ducing x rays observed as diagram lines therefore is
i
ni 	 n  (Li ) + n  (L N)(5)	 _	 .1
a
i
and the apparent diagram-line intensity is
I (L i
	 iX) = n. [R(LX)/T. ]	 (6)i 	 i
where R(LiX) is the radiative transition rate filling L i holes with X-shell
electrons, and Ti ' is the total transition rate, excluding Coster-Kronig
transitions. Similar expressions can be written for the satellite intensities.
i
it
6B. Numerical calculations
_ M
The Ag L1 Auger and Coster-Kronig transitions were recalculated using
Herman-Skillman 9,10 wave functions. The LZ L3M4 5 Coster-Kronig transitions
were calculated in intermediate coupling with the newly available transition
energies. 3,10 In Table IV we list the Coster-Kronig yields and L 1 widths for
two cases: (i) the full LL L 3M4 5 transition rate is included, (ii) the
,
calculated LL L3M4 5 transition rate is reduced by a factor of 1/2.5. The
Coster-Kronig yield f 23 is taken to be 0.143.11
The experimental L1 level width is found to be 3.95 eV by subtracting the
measured M2 3 level width 12 of 2.3 eV from the measured L63,4 (LZ M2,3 ) x-rayr
width  of 6.25 eV. It is seer: in Table IV that the "case (ii)" L 1 level width
agrees well with experiment, while the "case (i)" width that includes the
full calculated L1-L3M4 5 transition rate is too large, even though it repre-
r
sents an improvement over previous theoretical calculations. 13,14
The initial L-shell hole distribution is taken to be  N 1 :N 2 :N 3 =
1.00:1.48:2.83 assuming an incident Kramers spectrum. We follow the procedure
of Krause et al. 8 to find the Ag shakeoff probabilities P1=0.18, PiM=0.02,
and PiN=0.16, with i=1,2,3. The final hole distributions after Coster-Kronig
transitions, calculated with the above parameters and results, are listed in
Table V. The relative intensities (normalized to the apparent L3-M 5  diagram
line) are obtained by combining the final hole populations (Table V) and the
normalized radiative transition rates R(L 1.X)/T.' given by McGuire. I Resultsi
are listed in Table VI. For comparison, the experimental relative intensi-
ties 2 are also included in Table VI. Case (i) leads to an underestimate of
the Ll x-ray intensity and an overestimate of the strong S(L 3M4 5 ) satellite
r
.w.
i
.r
7
intensity by a factor of 2. Case (ii) leads to very good overall agreement
with experiment, except for underestimating the weak satellite lines S(L2M4)
and S(L2N4 ) by a factor of 2. This is probably due to an error in the shake-
off probability P 2M , since these satellites are produced by that shakeof£
process. If P 2 is taken to be 3% instead of 2% as used in the present
calculation, then the S(L2M4 ) and S(L2N4 ) satellite intensities are increased
by a factor of 1.5, bringing the result into much better agreement with ob-
servation.
IV. CONCLUSION
This analysis of the Ag L-series x-ray spectrum leads to several signifi-
cant conclusions.
Satellites arising from LN doubly-hole states are not, in general,
resolvable from the parent diagram lines. The observed high-energy satellite
lines are due to LM doubly ionized states created by LL L3M4 5 Coster-Kronig
transitions and by shakeoff. The observed peak structure in the satellite
spectrum is due to the multiplet splitting of the initial and final double-
hole states.
The L1 L3M4 5 Coster-Kronig transitions are energetically possible in
Ag. A L1 L3M4 5 transition-rate calculation using newly available Coster-
Kronig energies has reduced the discrepancy between theory and experiment,
but the theoretical result still is too large by a factor of ti2. it appears
that a reexamination of the theory of L 1 Coster-Kronig transitions is indicated.
L
i
i
a
3
31i3
z
a
)
1	 t
B
References
E. J. McGuire, Phys. Rev. 5 1 2313 (1972).
2L. G. Parratt, Phys. Rev. 54, 99 ( 1938).
3M. H. Chen, B. Crasemann, K.-N. Huang, M. Aoyagi,^,.%.:1 H. Mark, At. Data
Nucl. Data Tables 19, 0000 (1977).
4K.-N. Huang, M. Aoyagi, M. H. Chen, B. Crasemann, and H. Mark, At. Data
Nucl. Data Tables 18 1 243 (1976).
5J. A. Bearden, Rev. Mod. Phys. 39 1 78 (1967).
6E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge
U.P., Cambridge, England, 1953).
7F. P. Larkins, J. Phys. B: Atom. Molec. Phys. 9 1 37 (1976).
8M. O. Krause, F. Wuilleumier, and C. W. Nestor, Jr., Phys. Rev. A 6, 871
(1972).
9F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall,
Englewood Cliffs, N.J., 1963).
10M. H. Chen, B. Crasemann, K.-N. Huang, M. Aoyagi, and H. Mark, (unpublished).
11M. H. Chen and B. Crasemann, in Proceedings of the International Conference
on Inner-Shell Ionization Phenomena and Future Applications, ed..by R. W.
Fink, S. T. Manson, J. M. Palms, and P. Venugopala Rao. U.S. Atomic
Energy Commission Report No. CONF-720404 (unpublished), p. 43.
12L. I Yin, I. Adler, T. Tsang, M. H. Chen, D. A. Ringers, and B. Crasemann,
Phys. Rev. A 9, 1070 (1974).
13E. J. McGuire, Phys. Rev. A 3, 587 (1971).
148 Crasemann, M. H. Chen, and V. O. Kostroun, Phys. Rev. A 4 1 2161 (1971).
a
a
i
TABLE 2. Silver L x-ray energies (in eV).
Energy
Transition Experiment Theory
Parratta Beardenb
L1-N 3 3749.7
3749.8 3752.2
L1-N 2
3743.1 3743.2 3747.2
L
2-N 4 3519.5
3521.0
L1-M 5
3440.3 3439.2 3445.9
Ll M4 3432.2 3432.9 3439.9
L2
-N1 3428.6 3428.3 3421.4
L3 N5 3348.0 3348.8
L3-N1 3256.0 3256.0 3248.6
L1 M3 3234.7 3234.5 3238.2
L1-M 2
3203.6 3203.5 3207.0
LZ M4 3151.0 3150,.9 3151.7
L3-M5 2984.4 2984.3 2985.0
L3-M4 2978.3 2978.2 2978.9
LZ
 Ml 2805.5 2806.1 2803.4
,i
i
2633.8 2633.7 2630.6L3-Ml
aRef.	 2.
bRef.	 5.
i
f
V
1
j
I"
I10
TABLE II. Energy shifts (in eV) of x-ray transitions to the Ag L shell,
caused by the presence of an additional vacancy in the M or N shell. Results
from relativistic Hartree-Fork-Slater calculations.
State of spectator vacancy
Diagram line
M1	 M2	 M3
	
14	 M5	 N1	 N2	 N3	 N4	 N5
LZ M4 -	 -	 -	 -	 10.7 0.5 0.4	 0.7	 0.4 0.2
L3-M4 -	 -	 -	 -	 9.0
0.5 -	 0.5	 - 0.2
L3 M5 10.2	 11.4	 11.7	 11.1	 9.6 0.5 0.7	 0.5	 0.2 0.3
L3-N 5 30.2	 33.0	 30.0	 34.5	
32.6 5.2 4.6	 5.0	 2.1 1.9
11
TABLE III.	 Energies (in eV) of L3M4 5 M4 5M4 5 Lal 2 satellites, calculated
in intermediate coupling.
Energy
Transition
Theory Experiments
3P1 + 1S0
2977
Mixed_ with L3-M4 diagram
1P
+
1S0 2979 line at 2978.3 eV
1D2
 + 1D2 2981
1D2 + 3P1 2984
Mixed with L3-M5 diagram
3D3 + 1D2 985 line at 2984.4 eV
3D3 + 1G4 2986
3F4 + 3F3 2989
3D2 + 1D2 2990
3D3 + 3P2 2991
3P1 + 1D2 2991
2992
1D2 + 3F3 2991
1D2 + 3F2 2991
3D2 + 3P1 2992
TABLE III continued
Energy
Transition
Theory Experiment
3F4 -r 3F4 2994
1P1 + 
1D2 2994
3P1 + 3P1 2994
1F3 + 1D2 2995
3D	 r 3F 29953	 2 2995
3D3
 r 3F3 2995
3D2 r 3P2 2995
3P1 + 3P0 2995
3P0 + 3P1 2996
1F3 + 1G4 2996
1P1 + 3P1 2997
3P1 + 3P2 2997
1PY r 3P0 2998
3D2
 + 3F3 2999 2999
3D3 + 3F4 3000
3D2 + 3F2 3000
1P1 + 3P2 3000
1F3 ; 
3P2 3001
r s
12
L
T	 r
13
TABLE III continued
Energy
Transition
Theory	 Experiment
3P1	 ' 3F2 3002
ip1	 3F2 3005 3003
1F3 4' 3F3 3005
1F3 + 3F2 3006
3009
1F3	 3F4 3010
a Ref.	 2.
I14
TABLE IV.	 Silver LL level width P(L1) and Coster-Kronig yields.
Case	 f12 f 13 f 13 f13	 P(L1)	 (ev,)
(1)	 0.067 0.617 0.x13 0.730	 6.17
(ii)	 0.107 0.392 0.179 0.571	 3.89
a
z`
4
A
t
I	 1
15
3
TABLE V.
	 Final hole populations after Coster-Kronig transitions.
	 Case	 (i):
using full L1 
-L34,5 transition rate P(LZ L 3 M45 ).	 Case (ii):	 usingr
r(L1-L3M4 5)/2.5.r
n Case	 (i) Case	 (ii)
W
ni(L1 ) 0.166 0.264
n2 (L2 ) 1.040 1.040
n3 (L3 ) 2.321 2.321
nl (L1M) 0.006 0.006
n1 (L1N) 0.052 0.052
n2 (L2M) 0.0255 0.0255
n2 (L2N) 0.259 0.291
n 3 (L3M) 0.563 0.378
I	 I	 i	 I	 I'	 f9,
16
TABLE VI. Relative intensities of ng L-series x-ray lines.
Calculated relative intensity
Transition Experiment 
Case	 (i) Case	 (ii)
L1-M 2 2.62 3.73 3.9
M3 4.33 6.17 7.2
M4 0.03 0.04 0.04
M5 0.04 0.06 0.07
N2 0.49 0.69 0.66
N3 0.81 1.15 1.04
L2-M 1 1.42 1.43 2.0
M4 48.6 49.0 49.1
N1 0.28 0.28 0.28
M4 5.02 5.05 4.36
S(L2M4 ) 0.96 0.94 2.06
S(L2N4 ) 0.10 0.10 0.16
i	 L3-M 4.14 4.14 4.41
M4 11.3 11.3 8.61
M5 ` 100 100 100
N1 0.80 0..80 0.56
N5 10.1 10.1 11.9
S(L3M4 5 ) 20.6 13.6 11.7
S(L3N5 ) 1.87 1.36 1.78
aRef.	 2.
5
F
R
i


Interpretation of the silver L X-ray spectrum

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